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that we thus obtain a diverging lens, the curvature and action of which are variable.

23. Now let us suppose that we have increased the curvatures of the lens until the two surfaces nearly touch each other by their summits.* We might presume that if the removal of the liquid were continued, the mass would become disunited at that point at which this contact took place, and that the oil would recede in every direction towards the metallic band. This is, however, not the case; we then obscrve in the centre of the figure the formation of a small sharply defined circular space, through which objects no longer appear diminished, and we easily recognize that this minute space is occupied by a layer of oil with plane faces. If the removal of the liquid be gradually continued, this layer increases more and more in diameter, and may thus be extended to within a tolerably short distance of the solid surface. In my experiment, the diameter of the metallic cylinder was seven centimetres, and I have been enabled to increase the size of the layer until its circumference was not more than about five millimetres from the solid surface; but at this instant it broke, and the liquid of which it consisted rapidly receded towards that which still adhered to the metallic band. The fact which we have just described is very remarkable, both in itself and in the singular theoretical consequences to which it leads. In fact, that part of the mass to which the layer adheres by its margin presents concave surfaces, whilst those of the layer are plane; now the existence of such a system of surfaces in a continuous liquid mass seems in opposition to theory, since it appears evident that the pressures cannot be equal in this case. But let us investigate the question more minutely.

24. According to theory, the pressure corresponding to any point of the sur. face of a liquid mass, as we have seen, ($ 3,) is the integral of the pressures. exerted by each of the molecules composing a rectilinear line perpendicular to the surface at that point, and equal in length to the radius of the sphere of activity of the molecular attraction. The analytical expression of this integral contains no other variables than the radii of the greatest and of the least curva-. ture at the point under consideration, ($ 4,) consequently the pressure in question varies only with the curvatures of the surface at the same point. This, is rigorously true when the liquid is of any notable thickness; but we shall show that in the case of an extremely thin layer of liquid there is another element which exerts an influence upon the pressure. Let us conceive a liquid layer, the thickness of which is less than twice the radius of the sphere of sensible activity of the molecular attraction. Let each molecule be conceived to be the centre of a small sphere with this same radius, ($ 3,) and let us first; consider a molecule situated in the middle of the thickness of the layer. The little sphere, the centre of which is occupied by this molecule, will be intersected by the two surfaces of the layer, consequently it will not be entirely full of liquid; but the segments suppressed on the outside of the two surfaces being equal, the molecule will not be more attracted perpendicularly in one direction than in the other. Now let a small right line, normal to and terminating at the two surfaces, pass through this same molecule, and let us consider a second, molecule situated at some other point of this right line. The little sphere which belongs to the second molecule in question may again be intersected by the two surfaces of the layer; but then the two suppressed segments will be unequal; the molecule will consequently be subjected to a preponderating attraction, evidently directed towards the thickness of the layer. The molecule will then exert a pressure in this direction, and it must be remarked that this pressure will be less than if the liquid had any notable thickness, the moleculo

* To effect this operation, the point of the syringe must not be placed in the middle of the figure, as in the case of the doubly convex lens; but, on the contrary, near the metallic band, as this is now the point where the greatest thickness of the liquid exists.

being situated at the same distance from the surface; for in the latter case the little sphere would only be cut on one side, and its opposite part would be perfectly full of liquid. It might also happen that the little sphere belonging to the molecule in question in the thin layer is only cut on one side; the molecule will then still exert a pressure in the same direction, but its intensity will then be as great as in the case of a thick mass. It is easy to see that if the thickness of the layer is less than the simple length of the radius of the molecular attraction, the little spheres will all be cut on both sides; whilst if the thickness in question is comprised between the length of the above radius and twice this same length, a portion of the minute spheres will be cut on one side only. In both cases the pressure exerted by any molecule being always directed towards the middle of the thickness of the layer, it is evident that the integral pressure corresponding to any point of either of the two surfaces will be the result of the pressures individually exerted by each of those molecules, which, commencing at the point in question, are arranged upon half the length of the small perpendicular. Now each of the two halves of the small perpendicular being less than the radius of the sphere of activity of the molecular attraction, it follows that the number of molecules composing the line which exerts the integral pressure is less than in the case of a thick mass. Thus, on the one hand, the intensities of part or the whole of the elementary pressures composing the integral pressure will be less than in the case of a thick mass, and, on the other hand, the number of these elementary pressures will be less; from this it evidently follows that the integral pressure will be inferior to that which would occur in the case of a thick mass. P always denoting the pressure corresponding to any point of a plane surface belonging to a thick mass, (§ 4,) the pressure corresponding to any point of either of the surfaces of an extremely thin plane layer will therefore be less than P. Moreover, this pressure will be less in proportion as the layer is thinner, and it may thus diminish indefinitely; for it is clear that it would be reduced to zero if we supposed that the thickness of the layer was equal to no more than that of a simple molecule. We can obtain liquid layers with curved surfaces; soap-bubbles furnish an example of these, and we shall meet with others in the progress of this investigation. Now by supposing the thickness of such a layer to be less than twice the radius of the molecular attraction, we should thus evidently arrive at the conclusion that the corresponding pressures at either of its two surfaces would be inferior in intensity to those given by paragraph 4, and that, moreover, these intensities are less in proportion as the layer is smaller. We thus arrive at the following new principle: In the case of every liquid layer, the thickness of which is less than twice the radius of the sphere of activity of the molecular attraction, the pressure will not depend solely upon the curvatures of the surfaces, but will vary with the thickness of the layer. 25. We thus see that an extremely thin plane liquid layer, adhering by its edge to a thick mass the surfaces of which are concave, may form with this mass a system in a state of equilibrium; for we may always suppose the thickness of the layer to be of such value that the pressure corresponding to the plane surfaces of this layer is equal to that corresponding to the concave surfaces of the thick mass. Such a system is also very remarkable in respect to its form. inasmuch as surfaces of different nature, as concave and plane surfaces, succeed each other. This heterogeneity of form is, moreover, a natural consequence of the change which the law of pressures undergoes in passing from the thick to the thin part. 26. As we have already seen, theory demonstrates the possibility of the existence of such a system in a state of equilibrium. As regards the experiment which has led us to these considerations, although the result presented by it tends to realize in an absolute manner the theoretical result, there is one circumstance which is unfavorable to the completion of this realization. We can understand that the relative mobility of the molecules of oil is not sufficiently great to occasion the immediate formation of the liquid layer with that excessive tenuity which is requisite for equilibrium; the thickness of this layer, although very minute, absolutely speaking, is undoubtedly, during the first moments, a considerable multiple of the theoretical thickness. If, then, we produce the layer without extending it to that limit to which it is capable of increasing during the operation, and afterwards leave it to itself, the pressure corresponding to its plane surfaces will still exceed that o to the concave surfaces of the remainder of the liquid system. Hence it follows that the oil within the layer will be driven towards this other part of the system, and that the thickness of the layer will progressively diminish. The equilibrium of the figure will then be apparent only, and the layer will in reality be the seat of continual movements. The diminution in thickness, however, will be effected slowly, because in so confined a space the movements of the liquid are necessarily restrained; this is why, as in the experiment in paragraph 17, the mass only acquires its figure of equilibrium slowly, because there is a cause which impedes the movements of the liquid. The thickness of the layer gradually approximates to the theoretical value, from which the equilibrium of the system would result; but unfortunately it always happens that before attaining this point the layer breaks spontaneously. This effect depends, without doubt, upon the internal movements of which I have spoken above. We can imagine, in fact, that when the layer has become of extreme thinness, the slightest cause is sufficient to determine its rupture. The exact figure which corresponds to the equilibrium is therefore a limit towards which the figure produced tends; this limit the latter approaches very nearly, and would attain if it were not itself previously destroyed by an extraneous cause. Our experiment has led us to modify the results of theory in one particular instance; but we now see that, far from weakening the principles of this theory, it furnishes, on the contrary, incomplete as it is, a new and striking verification of it. The conversion of the doubly concave lens into a system comprising a thin layer is connected with an order of general facts: we shall see that a large number of our liquid figures become transformed, by the gradually produced diminution of the mass of which they are composed, into systems consisting of layers, or into the composition of which layers enter. 27. If by some modification of our last experiment we could succeed in ob-. taining the equilibrium of the liquid system, we might be able to deduce from it a result of great interest—an indication of the value of the radius of the sphere of activity of the molecular attraction. In fact, we might perhaps find out some method of determining the thickness of the layers; these might, for instance, then exhibit colors, the tint of which would lead us to this determination. Now we have seen that in the state of equilibrium of the figures, half the thickness of the layer would be less than the radius in question; hence we should then have a limit above which the value of this same radius would exist. In other words, we should know that the molecular attraction produces sensible effects, even at a distance from its centre of action beyond this limit. Our -experiment, although insufficient, may thus be considered as the first step towards the determination of the distance of sensible activity of the molecular attraction, of which distance at present we know nothing, except that it is of extreme minuteness. 2S. Let us now return to the consideration of thick masses. It follows from the experiments related in paragraphs 13, 14, 17, 18, and 21, that when a continuous portion of the surface of such a mass rests upon a circular periphery, this surface is always either of spherical curvature or plane. But to admit this oil. in all its generality, we must be able to deduce it from theory. We shall do this in the following series, at least on the supposition that the portion

of the surface in question is a surface of revolution. We shall then see that this same principle is of great importance. We may remark here that in the experiment in paragraph 23 the layer commences to appear as soon as the surfaces can no longer constitute spherical segments. Now we shall again find that in the other cases, when a full figure is converted, by the gradual withdrawal of the liquid, into a system composed of layers, or into the composition of which layers enter, the latter begin to be formed when the Fig. 3. figure of equilibrium, which the ordinary law of pressures would determine, ceases to be possible. The mass then assumes, or tends to assume, another figure, compatible with a modification of this law. Such is the general principle of the formation of layers under the circumstances in question. 29. After having formed a converging and a diverging liquid lens, it appeared to me curious to combine these two kinds of lens, so as to form a liquid telescope. For this purpose, I first substi. tuted for the ring of iron wire, in paragraph 18, a circular plate - of the same diameter, perforated by a large aperture. (Fig. 8) -o-o: This plate having been turned in alathe, I was certain of its being perfectly circular, which would be a very difficult condition to fulfil in the case of a simple curved iron wire. In the second place, I took for the solid part of the doubly concave lens a band of about two centimetres in breadth, and curved into a cylinder three and a half centimetres in diameter. These two systems were arranged as in Fig. 9, in such a manner that the entire apparatus being

suspended vertically in the alcoholic mixture by the iron wire a, and the two liquid lenses being formed, their two centres were at the same height, and ten centimetres distant from each other. In this arrangement the telescope cannot be adjusted by altering the distance between the objective and the eye-piece; but this end is attained by varying the curvatures of these two lenses. With the aid of a few preliminary experiments, I easily managed to obtain an excel. lent Galilean telescope, magnifying distant objects about twice, like a common opera-glass, and giving perfectly distinct images with very little irisation. b .." which represents a section of the system, shows the two lenses com: illed.

Figures of equilibrium terminated by plane surfaces. Liquid polyhedra. Laminar figures of equilibrium.

30. In the experiment detailed at paragraph 21, we obtained a figure presenting plane surfaces. These were two in number, parallel, and bounded by circular peripheries; but it is evident that these conditions are not necessary in order to allow plane surfaces to belong to a liquid mass in equilibrium. We can understand that the forms of the solid contours might be indifferent, provided they constitute plane figures. We can, moreover, understand that the number and the relative directions of the plane surfaces may be a matter of indifference, because these circumstances exert no influence upon the pressures which correspond to these surfaces, pressures which will always remain equal to each other. Lastly, it follows from the principle at which we arrived at the end of paragraph 20, relative to the influence of solid wires, that for the establishment of the transition between a plane and any other surface, a metallic thread representing the edge of the angle of intersection of these two surfaces will be sufficient. We are thus led to the curious result, that we ought to be able to form polyhedra, which are entirely liquid excepting at their edges. Now, this is completely verified by experiment. If for the solid system we take a framework of iron wire representing all the edges of any polyhedron, and we cause a mass of oil of the proper volume to adhere to this framework, we obtain, in fact, in a perfect manner, the polyhedron in question; and the curious spectacle is thus obtained of parallelopipedons, prisms, &c., composed of oil, and the only solid part of which is their o: To produce the adhesion of the liquid mass to the entire framework, a volume is first given to the mass slightly larger than that of the polyhedron which it is to form; it is then placed in the framework; and, lastly, by means of the iron spatula, (§ 9,) which must be introduced by the second aperture of the lid of the vessel, and which is made to penetrate the mass, the latter is readily made to attach itself successively to the entire length of each of the solid edges. The excess of oil is then gradually removed with the syringe, and all the surfaces thus become simultaneously exactly plane. But that this end may be attained in a complete manner, it is clearly requisite that the equilibrium of density between the oil and the alcoholic mixture should be perfectly established; and the slightest difference in this respect is sufficient to alter the surfaces sensibly. It should also be borne in mind that the manipulation with the spatula sometimes occasions the introduc- t tion of alcoholic bubbles into the interior of the mass of Fig ZZ oil. These are, however, easily removed by means of the syringe. 31. Now, having formed a polyhedron, let us see what will happen if we gradually remove some of the liquid. Let us take, for instance, the cube, the solid framework of which, with its suspending wire, is represented at Fig.11.” Let the point of the syringe be applied near the middle of one of the faces, and let a small quantity of the oil be drawn up. All the faces will immediately become depressed simultaneously and to the same extent, so that the superficial square contours will form the bases of six similar hollow figures. We ..". imagined this to have been the case for the maintenance of equality between the pressures. If fresh portions of the liquid are removed, the faces will become more and more hollowed; but to understand what happens when this manipulation is continued, we must here enunciate a preliminary proposition. Suppose that a square plate of iron, the sides of which are of the same length as the edges of the metallic frame, is introduced into the vessel, and that a mass of oil equal in volume to that which is lost by one of the faces of the cube is placed in contact with one of the faces of this plate; I say that the liquid, after having become extended upon the plate, will present in relief the same figure as the

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* The edges of all the frames which I used were 7 centimetres in length.

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